The Third Asia-Pacific Conference on Combustion
June 24-27, 2001, Seoul, Korea
TIME SCALE ANALYSIS FOR OPPOSED-FLOW FLAME SPREAD - THE FOUNDATIONS
Subrata Bhattacharjee
Department of Mechanical Engineering
San Diego State University
San Diego, CA 92182, USA
Subrata@voyager5.sdsu.edu
ABSTRACT
Time scales: Figure 1 depicts a schematic of steady opposed-flow
flame spread in the flame fixed coordinates. The opposing flow
Opposed-flow flame spread over solid fuels has been extensively
studied over the last three decades. These studies have addressed
major regimes such as thermal regime with sub-regimes of thick
and thin fuels, kinetic regime, downward spread and microgravity
regimes among others. A general study, where all these
sub-regimes are defined and put into perspective, is still lacking.
In this paper, a novel scale analysis is presented which brings
together all different regimes and sub-regimes of opposed-flow
flame spread.
velocity V g , can be due to forced-flow or buoyancy induced flow.
With respect to the flame the oxidizer, assumed to be a mixture of
oxygen and nitrogen, approaches the flame with a velocity
Vr  V g  V f
and the fuel with a velocity
Vf
. To identify the
relevant time and length scales, attention is focussed on the
leading edge of the flame where the forward heat transfer, the
fundamental mechanism of any flame spread[2], occurs. Two
control volumes, one in the gas phase of size
The thermal regime, the backbone of this opposed-flow flame
spread problem, is first established through a time scale analysis in
which competition among different heat transfer mechanisms,
gas-phase and solid-phase kinetics are reduced to comparing
individual time scales expressed in terms of known parameters of
the problem. Various sub-regimes are established by removing
thermal-regime assumptions one at a time, and defining the
appropriate non-dimensional numbers through ratios of
competing times. Results from existing analytical solutions,
comprehensive computational model and available experimental
data are used to validate some of the conclusions and lay down the
foundations of scaling for the opposed flow flame spread
phenomena.
one in the solid phase of size
Lgx xLgy xW
and
Lsx xLsy xW , are drawn at the
W being the fuel-width in the z direction
and the length scales, L gx , L gy , L sx and Lsy , unknown at this
flame leading edge,
point.
The essence of flame spread mechanism[2], steady forward heat
transfer to the unburned fuel, with respect to these control
volumes of Fig. 1(a) can be stated as follows. In the gas-phase
control volume the vaporized fuel and oxidizer react to raise the
gas temperature from its ambient value
flame temperature
Keywords: flame spread, scale analysis, microgravity.
Tf
T
to a characteristic
. The velocity at which the solid fuel must
be fed through the solid-phase control volume in the negative x
direction to steadily raise the fuel temperature from its ambient
INTRODUCTION
value
Opposed-flow flame spread over solid fuels has been extensively
studied over the last three decades[1]. While most of the work
have been concentrated into separate studies of the sub-categories
such as the thermal regime, thick fuels, thin fuels, super-thin fuels,
downward spread, kinetic regime, radiative regime, and, recently,
microgravity regime, efforts to bring them together through a
single analysis have been relatively few[2]. Although scaling
arguments[2,3] have been used in the past to understand the
mechanism of flame spread and the complicated energy balance
on the fuel surface, in this work we offer a novel time scale
analysis encompassing all sub-regimes for the first time following
the scale-analysis conventions established by Bejan[4]. Existing
analytical solutions, comprehensive computational model and
available experimental data will be used to validate the results and,
whenever possible, determine critical values of the
non-dimensional numbers arising out of scaling arguments that
delineate the sub-regimes of opposed-flow flame spread.
TF , 
to a characteristic vaporization temperature
Tv
is the
desired spread rate V f . At any other velocity of the fuel the
location of the pyrolysis front, and hence the flame leading edge,
will not remain stationary rendering the problem unsteady.
For the flame to be established at the end of the gas-phase control
volume, combustion reaction must be complete within the
available residence time
t res, g ~
Lgx
. In terms of the
Vr
characteristic time for combustion t comb , this implies that t res , g
must be greater than
the other hand,
t res , g
t comb
for kinetics to be considered fast. On
must be small compared to
t ger
(see
discussions below and Fig. 1b) for radiative losses from the gas
1
phase to have no effect on
Tf
Solid-Forward conduction as the only surviving driving forces. It
has been experimentally and computationally verified that, of the
two Gas-to-Surface conduction is the dominant one in the thermal
. Therefore, finite-rate kinetics and
gas-phase radiative losses can be considered marginal allowing
the use of adiabatic flame temperature
T f ,ad for T f
if
regime, i.e.
t comb  t res, g  t ger .
Within the available residence time
t res, s ~ t sh  t vap ~ t sh ,
t res,s
L
~ sx
Vf
t sh ~ min( t gsc , t sfc , t gsr , t esr ) ~ t gsc ,
in the solid
TF , 
t res , s ~ t gsc
therefore,
phase control volume, the fuel must be preheated and part of it
vaporized. That is:
where
Combining these assumptions, Eq. (1)
can be simplified as follows.
Under this simplifying scenario the gas phase allows the flame to
spread as fast as it can meet the requirement of the solid phase.
t res,s ~ t sh  tvap ,
t sfc  t gsc .
(3)
What remains is to be done is to express these time scales in terms
of known parameters of the problem before Eq. (3) can be solved
to obtain the flame spread rate under the restricting constraints of
the thermal regime.
(1)
Length Scales: In the gas-phase, a balance between the
conduction and convection in the x-direction at the leading edge
t sh is the time for sensible heating of the solid phase from
to Tv and t vap is the characteristic time for the pyrolysis
yields the familiar5 expression for
Vr T f  T 
of fuel.
The characteristic time for a given heat transfer mechanism is
defined as the time required by that mechanism, acting alone, to
supply the necessary heat to perform the most basic task, that is to
preheat the solid-phase control volume. The fastest pathway, i.e.
the mechanism with the smallest time scale, is, clearly, the
dominant driving force behind the spread. Figure 1(c) depicts the
competing heat transfer mechanisms during the solid-phase
Lgx
or,
Lgx ~
~
L gx .
 g T f  T 
L2gx
g
(4).
Vr
residence time. Conduction from Gas-to-Solid ( t gsc ),
The transverse length scale
Solid-Forward conduction ( t sfc ), radiation feedback from
Delichatsios[3] as the diffusion length in the y-direction within the
available residence time.
Gas-to-Solid ( t gsr ), and Environment-to-Solid radiation, possibly
t res, g ~
from an external source or a large downstream plume, if any, ( t esr )
act in parallel to supply the sensible and latent heat to the virgin
fuel. The radiation loss from the Solid to-Environment ( t ser ), on
Vr

g
or,
Vr  V g
The pyrolysis chemistry is considered fast, i.e.
g
Vr
 Lgx ,
(5)
The gas-phase conduction being the driving force,
on the solid phase making
Lsx ~ Lg .
Lg
is imposed
The transverse length
L sy , derived in a manner similar to L gy , however, cannot be
greater than the half-thickness of the fuel. Therefore,
t comb  t res, g  t ger
V g  V f
,
Vr2
Lg  Lgx  Lgy
t res, s .
Thermal Regime: The thermal regime of flame spread is defined
with the following simplifying assumptions. In the gas phase,
radiation loss in the gas-phase is assumed negligible, while
combustion kinetics is considered fast. Moreover, the opposing
flow velocity is considered much greater than the spread rate.
while,
can be obtained following
Lgy ~  g t res, g 
the other hand, is an interfering mechanism, just as the radiative
losses in the gas phase control volume, its importance dependent
on how t ser compares with
Lgx
L gy
(2)
Lsx ~ Lg 
t vap ~ 0 , so that
g
Vr
and,


 s g 
L 
Lsy ~ min  ,  s sx   min  ,
.
V
V
V

f 
f r 




the solid-residence time is spent almost entirely on the sensible
heating through the fastest heating mechanism. If radiation is
entirely neglected, this leaves Gas-to-Surface conduction and
2
(6).
Spread Rate Formulas in the Thermal Regime: The amount of
heat transfer necessary to preheat the solid-phase control volume
from
TF , 
to the vaporization temperature

Tv ,
Qchar ~  s c s Lsy LsxW Tv  TF ,  , represents a

 s c s Lsy LsxW Tv  TF , 
Qchar
~
T f  Tv 
 L gxW
q gsc
g
L gxW
L gy
F
 s c s Lsy
T f  Tv
Tv  TF ,
, and
F,
g
F
 s c s
g  g cg 2
~ Vr
F ,
 s  s c s
V f ,thick
(11)
1.
Di Blasi, C., Prof. Energy and Combust. Sci., Vol. 19,
pp. 71-104. (1993)
2.
Williams, F.A., Sixteenth Symposium (International) on
Combustion, The Combustion Institute, Pittsburgh, PA,
p. 1281, (1976).
3.
Delichatsios, M.A. Twnety-Sixth Symposium
(International) on Combustion, The Combustion
Institute, Pittsburgh, PA, p. 1281, (1996)..
4.
Bejan, A., Convection Heat Transfer, John Wiley and
Sons, (1995)
5.
de Ris, J.N., Twelfth Symposium (International) on
Combustion, The Combustion Institute, Pittsburgh, PA,
p. 241, (1969).
6.
Delichatsios, M.A., Combust. Sci. and Tech., Vol. 44,
pp. 257-267, (1986).
7.
Bhattacharjee, S., King, M., Nagumo, T, Takahashi, S.,
and Wakai K, Proc. Combust. Inst. 28: in press.
.(8)
L sy in Eq. (8) produce:
V f ,thin ~
,
References:
where,
  Lg 
Lsy ~ min  , s

  g F 
The thin and thick limits of
 cr ,thinthick, EST
Conclusions: A general opposed-flow flame spread problem is
studied here with the help of a novel time-scale analysis. Based on
the foundations presented here, the thick and thin-fuel
sub-regimes are combined into a single unified formula (Eq. 11).
The analysis can be extended to include various other regimes of
flame spread, a topic left for a future presentation.
(7)
Substituting Eq. (7) into Eqs. (3) and (6) results in:
Vf ~

Spread rates computed with a comprehensive numerical model
have validated this formula, at least for the downward
configuration7. Computational results also suggest that the fuel
becomes thermally thick for T  2 , an apparently universal
conclusion.
Lsy  s c s 1
Vr  g c g F
g
T
 1
 min 1, 
 T
is, therefore,
 , alone, to supply Qchar .
q gsc
is the time taken by
t gsc ~
t gsc
V f ,thick, EST
where,
characteristic amount of energy at the heart of the flame spread
mechanism. The gas-to-solid conduction time
Vf
and
(9)
These expressions are identical (except for a constant  / 4 for
thin fuel) to the analytical solutions of de Ris[5] and
Delichatsios[6] indicating the strength of this simplified scaling
approach to this complex problem.
ACKNOWLEDGEMENT
Thick vs. Thin Fuels: Although the transition between the thin
and the thick limits cannot occur abruptly, the critical thickness
obtained by equating the spread rate expressions in the two limits
[Eq. 9] should provide, within an undetermined constant, the
vicinity around which the transition takes place.
 cr ,thinthick ~
 s Lg
g F
.
This work was supported by NASA (Contract NCC3-842) Glenn
Research Center with Dr. Sandra Olson serving as the contract
monitor.
(10)
If the fuel half-thickness is non-dimensionalzed with such an
expression, the spread rate formulas in the thick and the thin
regimes can be combined.
3
y
Lgx
Vr  Vg  V f
gx
Lgy
x
Lsy
Lsx
t ser
t gsr
t ger
Environment (e)
esr
tesr
t sfc
tcomb
t res, g 
Vf

gsr
t gsc
Lgx
Vr
t vap
t sh
tres , s 
Lsx
Vf
gsc
Gas (g)
sfc
Solid (s)
Fig. 1. (a) Control volumes in the gas and solid phases at the leading edge. (b) Available time
which combustion must occur (blue). (c) Available time
ser
t res, s
t res , g
(black) in the gas phase in
in the solid phase (black) in which sensible heating and
vaporization must occur (blue) through competing driving mechanisms (red). The green clocks represent non-essential parallel
processes. (d) Close up of the leading edge illustrating the various heat transfer pathways.
1